New York Journal of Mathematics
New York J. Math. 25(2019) 1421–1437.
Abelian Hopf Galois structures from almost trivial commutative
nilpotent algebras
Lindsay N. Childs
Abstract. LetL/K be a Galois extension of fields with Galois group Gan elementary abelian p-group of rankn for pan odd prime. It is known that nilpotent Fp-algebra structures A onG yield regular sub- groups of the holomorph Hol(G), hence Hopf Galois structures onL/K.
In this paper we illustrate the richness of Hopf Galois structures on L/Kby examining the case whereAis abelian ofFp-dimensionnwhere dim(A2) = 1. We determine the number of Hopf Galois structures that arise in these cases, describe those structures explicitly, and estimate the extent of failure of surjectivity of the Galois correspondence for those structures.
Contents
1. Introduction 1421
2. Hopf Galois structures from nilpotent algebras 1423
3. Working in the affine group 1425
4. The caser =n−1 1426
5. The number of Hopf Galois structures associated to A 1427
6. The cases n= 2,3,4 1431
7. The Galois correspondence ratio 1434
References 1436
1. Introduction
In 1969, Chase and Sweedler [5] defined the notion of a Hopf Galois exten- sion of fields by abstracting the formal properties of a classical Galois exten- sion of fields. In 1987, Greither and Pareigis [13] discovered that a classical Galois extension L/K of fields with Galois group G could also be a Hopf Galois extension for a K-Hopf algebra H other than H = KG, the group ring of the Galois group, acting in the obvious way onL. They showed that
Received August 6, 2019.
1991Mathematics Subject Classification. 13B05, 12F10, 20B35, 13M05.
Key words and phrases. Finite commutative nilpotent algebras, abelian Hopf Galois extensions of fields, regular subgroups of finite affine groups.
ISSN 1076-9803/2019
1421
LINDSAY CHILDS
determining the number of Hopf Galois structures onL/K depends solely on the Galois group G. More precisely, the Hopf Galois structures correspond to regular subgroupsN of the permutation group of Gthat are normalized by the image λ(G) of the left regular representation of G in Perm(G). In many examples the subgroup N of Perm(G) need not be isomorphic to G, so we define thetype of the Hopf Galois extension ofL/K corresponding to N to be the isomorphism class of the group N.
Since the appearance of [13] there has been a fairly steady sequence of papers studying the number of Hopf Galois structures on a Galois extension of fields L/K with Galois group G. These range from Byott’s uniqueness paper [2] and his theorem [3] that if G is a non-abelian simple group then L/K has exactly two Hopf Galois structures, to papers that for suitable Galois groupsGdescribe large numbers of Hopf Galois structures on L/K, e.g. [7], or describe Hopf Galois structures of all possible types, e. g. [1].
Counting Hopf Galois structures on a field extension with Galois groupG is often made easier by translating the problem of finding regular subgroups of Perm(G) that are isomorphic to a given groupN and are normalized by λ(G) to a problem of finding regular subgroups of Hol(N) that are isomor- phic to G. This translation from Perm(G) to Hol(N) was first codified in [2], and has been the approach of choice for most papers devoted to counting Hopf Galois structures.
In [4] and subsequently in [12] Caranti, et. al. showed that for a finite abelian p-group, any commutative regular subgroup of Hol(G) can be ob- tained as the circle, or adjoint, group of a commutative nilpotent algebra structure (G,+,·) on the additive groupG. Meanwhile, Rump [15] defined a left brace and showed that if (A,+,·) is a radical ring, then (A,◦,+) is a left brace, and Guarneri and Vendramin [14] extended the concept to that of a skew left brace by relaxing the commutativity assumption on the operation +. In particular, they characterized skew braces as follows:
Theorem 1.1. Let (B,◦, ?)be a set with two group operations◦ and?. Let λ◦, λ? : B → Perm(B) be the two left regular representation maps, defined by λ◦(b)(x) = b◦x, λ?(b)(x) = b ? x. Let Hol(B, ?) ⊂ Perm(B) be the normalizer of λ?(B) in Perm(B). Then B is a skew left brace if and only if λ◦(B)⊂Hol(B, ?).
Subsequently, Byott and Vendramin [16] showed that (B,◦, ?) is a skew left brace if and only if there exists a Galois extension L/K with Galois group G and a Hopf Galois structure of type N so that G ∼= (B,◦) and N ∼= (B, ?). Their observation follows from the fact from Greither-Pareigis that if N = (B, ?) is normalized by λ◦(B) = λ(G) in Perm(G), then N corresponds to a Hopf Galois structure on L/K, and conversely, if L/K is G-Galois and has a Hopf Galois structure corresponding to the regular subgroup N of Perm(G) normalized by λ◦(G) in Perm(G), then N defines a new group structure (G, ?) ∼=N on G which makes (G,◦, ?) into a skew brace.
To illustrate the richness of Hopf Galois structures, especially on Galois extensions with Galois group an elementary abelian p-group, insights can be gained by just looking at those arising from nilpotent Fp-algebras. In this paper we look at Hopf Galois structures corresponding to a class of commutative nilpotent Fp-algebras A of Fp-dimension n that are “almost”
trivial. Thus we assume that A has the property that dimFp(A2) = 1 and A3 = 0. If A2 = 0, then the Hopf Galois structure on a Galois extension with Galois group G ∼= (A,+) is unique, namely that given by the Galois group, so our examples of nilpotent algebras are about as close to being trivial as possible.
We determine the isomorphism types of commutative nilpotentFp-algebras Aof dimensionnwithA3 = 0 and dim(A2) = 1, and determine the number of regular subgroups of Hol(G) associated to each isomorphism type. For n = 4 this approach yields more than p9 regular subgroups. We describe the Hopf Galois structure on L/K corresponding to each regular subgroup arising from a given isomorphism type of algebra. We also explicitly de- scribe the Hopf algebra action on L/K, and estimate the extent of failure of the Galois correspondence for the Hopf Galois structure to map onto the intermediate fields betweenK andL.
Throughout, let L/K be a Galois extension of fields with Galois group Γ, an elementary abelian p-group of orderpn, p an odd prime. For a finite abelian p-group G with operation +, a ring structure A = (G,+,·) on G will be called nilpotent if A is associative and nilpotent: Am = 0 for some m >1.
This research was inspired by discussions with Tim Kohl. Many thanks to him for sharing his enthusiasm with me. My thanks also to the referee for some insightful comments.
2. Hopf Galois structures from nilpotent algebras
LetA= (A,+,·) be a nilpotentFp-algebra ofFp-dimension n. The circle operation ◦ on A, defined by a◦b = a+b+a·b for a, b in A, is clearly associative with identity element 0. Since A is nilpotent, the circle inverse aof ais
a=−a+a2−a3+. . . ,
where ar = a·a·. . .·a (r factors), and so (A,◦) is a group, the adjoint group of A. It is well known since [15] that then (A,◦,+) is a left brace with additive group (A,+), that is, for all a, b, cinA
a◦(b+c) =a◦b−a+a◦c.
LetAn={a1·a2·. . .·an:a1, . . . , anin A}. Then we have
Proposition 2.1. Let (A,+) be an abelian p-group of finite order pn, and let (A,+,·) be a nilpotent ring structure on(A,+). Let(A,◦) be the adjoint group on A and let λ+, λ◦ be the corresponding left regular representations of A into Perm(A). Then the following are equivalent:
LINDSAY CHILDS
(i) λ◦(A) is normalized by λ+(A).
(ii) A3 = 0.
(iii) (A,◦,+) is a left skew brace with(A,◦) acting as the additive group.
In [9] a skew brace satisfying (iii) was called a bi-skew brace.
The equivalence of (i) and (iii) follows by the Guarneri-Vendramin charac- terization, Theorem1.1above. The equivalence of (ii) and (iii) is Proposition 4.1 of [9], a routine computation working moduloA4. As the referee kindly pointed out, the equivalence of (i) and (ii) for finite dimensional nilpotent algebras over a field was observed immediately following Lemma 3 in [4].
We identify the corresponding Hopf Galois structures:
Corollary 2.2. Let L/K be a Galois extension with Galois group (A,+) = G, an abelian p-group of order pn. Let A = (A,+,·) be a commutative nilpotent ring structure on (A,+) and suppose A3 = 0. Then T =λ◦(A)⊂ Perm(A) yields a Hopf Galois structure on L/K by a K-Hopf algebra H, where
i) H is the fixed ring of LT under the action of G:
H=LTG = (
X
x∈G
bxλ◦(x) :bx−x·z =bzx for all z in G )
; ii) H acts on L by
X
x∈G
bxλ◦(x)
!
(a) =X
x∈G
bxa−x+x2 for b, a in L.
Proof. Let {ez : z ∈ G} be the dual basis to the basis G = (A,+) of the group ringL[G]. The action ofT =λ◦(A) on GL=P
z∈GLez is by λ◦(x)(ez) =ex◦z
for x in G. This yields an action of the group ring LT on GL making GL an LT-Hopf Galois extension ofL. Sinceλ+(G) acts on T by
λ+(z)λ◦(x)λ+(−z) =λ◦(x−x·z), the correspondingK-Hopf algebra is
H =LTG= (
X
x∈G
bxλ◦(x) :bx−x·z=bzx for all z inG )
where forainL andy inG,ay is the image ofaunder the Galois action of y on L, and H acts on GLby
X
x∈G
bxλ◦(x)
!
X
y∈G
ayey
= X
x,y∈G
bxayex◦y.
NowL embeds inGL by
a7→ X
y∈G
ayey. So the action of H on Lis by
X
x∈G
bxλ◦(x)
!
(a) = X
x,y∈G,x◦y=0
bxay.
Sincex3= 0,x◦(−x+x2) = 0, and so the action ofH on Lcan be written X
x∈G
bxλ◦(x)
!
(a) =X
x∈G
bxa−x+x2.
There is no a priori reason why (A,◦) and hence T = λ◦(A) should be isomorphic toG, so thatH has typeG. But if Ais commutative, it is true:
we note the following variant of Theorem 1 of [12]:
Proposition 2.3. Let p > 3 be an odd prime and G = (G,+) be a finite abelian p-group of order pn. Let A = (G,+,·) be a commutative nilpotent ring structure on (G,+) and suppose A3 = 0. Then the regular subgroup N = (G,◦) of Hol(G)⊂Perm(G) is isomorphic to (G,+).
The statement of Theorem 1 of [12] replaces the condition A3 = 0 in Proposition 2.3 by the condition that the p-rank m of G should satisfy m+ 1 < p. The proof of Proposition 2.3 is essentially the same as that of Theorem 1 of [12]. The only change is that the condition A3 = 0 implies that ap = 0 for all a in A, which slightly simplifies the proof in [12]] by eliminating the need to apply a condition on thep-rank of Gto insure that ap does not interfere with the induction argument.
3. Working in the affine group
For the remainder of the paper we restrictGto be an elementary abelian p-group of p-rank n >1, and we consider only commutative nilpotent ring structures Aon (G,+) with A3 = 0. From [7], it is known that the number of isomorphism types of such structures is bounded from below bypb where b=O(n3).
Each such ring is anFp-algebra. Let dimFpA/A2 =rand dimFpA2 =n−r.
Given an Fp-basis (x1, . . . , xr) of A/A2 and a basis (y1, . . . yn−r) of A2, the multiplicative structure ofAwith those bases is given by a set Φ(k) = (φ(k)ij ) of r×r symmetric matrices with coefficients inFp, by the equations
xixj =
n−r
X
k=1
φ(k)i,jyk.
LINDSAY CHILDS
The group structure (G,◦) onGarising from (A,+,·) depends on the matri- ces{Φ(k)}, and hence so does the regular subgroupT =λ◦(G) of Perm(G).
It is convenient to view Hol(G) as the affine group Affn(Fp) and realize the regular subgroup T inside Affn(Fp).
Let Affn(Fp) be the subset of GLn+1(Fp) consisting of matrices of the form
B v 0 1
,
whereB is ann×nmatrix, v is a column vector inFnp, 0 is a n-row vector of zeros and 1 is a 1×1 identity matrix. Then Affn(Fp) may be identified as the holomorph Hol(Fnp) =λ(Fnp)·Aut(Fnp) of the additive group Fnp, where the matrices
P 0 0 1
withP in GLn(Fp) form the subgroup Aut(Fnp) of Hol(Fnp), and matrices I x
0 1
forx inFnp form the subgroup λ(Fnp).
The groupT =λ◦(A) embeds as a regular subgroup of Affn(Fp) as follows:
The map λ◦ from Fnp toFnp is given by λ◦(x)(y) =x◦y =x+y+x·y.
Write λ◦(x)(y) =x+y+Lx(y), where Lx(y) = x·y. Then Lx is a linear function from Fnp to Fnp, so has a matrix relative to the standard basis of Fnp that we also callLx. Then λ◦(x) in Affn(Fp) becomes then+ 1×n+ 1 matrix
Tx =
I+Lx x
0 1
, because for any y inFnp, we have
Tx
y 1
=
y+Lx(y) +x 1
=
y+x·y+x 1
=
λ◦(x)(y) 1
.
4. The case r = n−1
We now look at the class of examples where dim(A) = n,dim(A2) = 1, A3 = 0. Then A has the Fp-basis (x1, . . . , xn−1, xn) withxixj =φi,jxn, so A is determined by that basis and the single n×n structure matrix Φ = (φij). Since A3 = 0, φni = φin = 0 for all i. In this section we determine the regular subgroups of Affn(Fp) associated to A.
Since Ais commutative, the structure matrix Φ is symmetric. Then (c.f.
[17], Section 3.4.6) there is an invertible matrix P so that PΦPT = D = diag(Ds,0) is diagonal, whereDs= diag(1, . . . ,1, s) isk×kfor somek≤n, wheresis either 1 or any chosen non-square in Fp.
So setz=P x. Thenzn=xnand with respect to the basis (z1, . . . , zn−1, zn), A has the structure matrix Dwithzizj =dijzn and
D= diag(d1, . . . , dn) = diag(1,1, . . . ,1, s,0, . . .0) withs=dk.
So by that change of basis ofA, we can realize the groupT conveniently in
Hol(G)∼= Affn(Fp) =
GLn(Fp) Fnp
0 1
. by picking the basis (z1, . . . , zn) for A so that Φ =D.
Let {e1, . . . , en} be the standard basis of Fnp corresponding to the basis {z1, . . . zn} ofA= (A,+). Then λ◦(zi) =Ti is the element
Ti=
Li ei
0 1
, which acts on A = {r =Pn
i=1riei :r ∈ Fnp} embedded as elements r1 in Fn+1p by
Li ei
0 1
ej
1
=
ei◦ej
1
=
ei+ej
1
fori6=j Li ei
0 1
ei 1
=
ei◦ei 1
=
ei+ei+dien 1
.
5. The number of Hopf Galois structures associated to A In this section we determine the number of Hopf Galois structures on a Galois extension L/K of fields with Galois group G = (Fnp,+) that corre- spond to certain isomorphism types of nilpotent algebra structures on G.
To do so, we have
Proposition 5.1. Let A be a nilpotent Fp-algebra structure on (Fnp,+).
Then the number of Hopf Galois structures on L/K corresponding to the isomorphism type of A is equal to
|GLn(Fp)|/|Sta(T)|
whereT =λ◦(A) is the regular subgroup ofAffn(Fp)corresponding to Aand Sta(T) ={P ∈GLn(Fp) :
P 0 0 1
T =T
P 0 0 1
.
This follows by [4], which showed that two nilpotentFp-algebras on (Fnp,+) are isomorphic if and only if the corresponding regular subgroups of Affn(Fp) are conjugate by an element of Aut(G) =
GLn(Fp) 0
0 1
in Affn(Fp).
We note that given a regular subgroup T of Affn(Fp) normalized by λ+(A), corresponding to A withA3 = 0, then all of the regular subgroups in the orbit of T under conjugation by Aut(G) correspond to algebras A1
LINDSAY CHILDS
isomorphic to A, hence have A31 = 0. Thus all are normalized by λ(G).
Hence by Galois descent, all of those regular subgroups give rise to Hopf Galois structures on a Galois extension L/K with Galois group G.
The commutative nilpotentFp-algebraAwithA2= 0 yields the classical Galois structure on a Galois extension with Galois group G. For then Φ = D = 0, and the corresponding regular subgroup T of Affn(Fp) is λ+(A), which is stable under conjugation by every element of Aut(G), hence yields only the classical Galois structure onL/K.
Since pis odd, we may assume that Φ = diag(Ds,0). Let vs= (r1, r2, . . . , rk−1, srk)T,
v= (r1, r2, . . . , rk−1, rk)T, w= (rk+1, . . . , rn−1)T
(column vectors of elements ofFp). Then it is convenient to write elements of T =λ◦(A) as block matrices of the form
T ={λ◦(r) =
I 0 0 v
0 I 0 w
vTs 0 1 rn
0 0 0 1
:r ∈Fnp}
where the diagonal entries are identity matrices of sizek×k, (n−1−k)× (n−1−k), 1×1 and 1×1, respectively.
To determine the number of Hopf Galois structures corresponding to reg- ular subgroups in the orbit of T, we need to find the stabilizer of T under conjugation by the elements of Aut(G) = GLn(Fp).
To determine the stabilizer of T, we seek the set of (n+ 1)×(n+ 1) matrices
Q=
P 0 0 1
in Aut(G) ⊂ Affn(G) so that QT Q−1 = T, where P in GLn(Fp) has the form
P =
P11 P12 P13
P21 P22 P23 P31 P32 P33
with blocks of the same size as λ◦(r). Letλ◦(r0) be another element of T.
We compute P λ◦(r) and λ◦(r0)P and set P λ◦(r) =λ◦(r0)P. Equating the (11) terms yields thatP13= 0.
Equating the (21) terms yields thatP23= 0.
Equating the (32) terms yields thatP12= 0.
Then equating the (31) terms yields
v0Ts P11=P33vsT. Equating the (14) terms yields
P11v=v0.
Equating the (24) terms yields
w0=P21v+P22w.
Equating the (34) terms yields
t0 =P31v+P32w+P33t.
The (24) and (34) equations definew0 andt0. SettingP33=q, a non-zero element ofFp, then from (14) and (31) we have
P11Tvs0 =qvs and P11v=v0.
Recalling thatDs= diag(1, . . .1, s), a k×kmatrix, then Dsv=vs,Dsv0= vs0. So
P11TDsv0=qDsv, hence
P11TDsP11v=qDsv.
ThusP is in the stabilizer ofT if P =
P11 0 0 0 P21 P22 0 0 P31 P32 P33 0
0 0 0 1
where
P33=q is in GL1(Fp);
P32 is 1×(n−1−k) and arbitrary;
P31 is 1×kand arbitrary;
P22 is in GLn−1−k(Fp);
P21 is (n−1−k)×kand is arbitrary; and P11 is in GLk(Fp) and satisfies P11TDsP11=qDs.
As noted above, we may assume that A has a basis for which A has the structure matrixD=
Ds 0
0 0
whereDs= diag(1,1, . . . , s) isk×k,k < n.
To determine the possibleP11 we have three cases:
(1)k is odd;
(2)k is even ands= 1
(3)k is even andsis a non-square in Fp.
Each case involves a different orthogonal group. The notation for the orthogonal groups over Fp is from [17], Section 3.7.
Proposition 5.2. For Case 1), let k= 2m+ 1. For all q 6= 0in Fp, there exists ak×kmatrixC so thatCTC =qI if and only ifq is a square. Fixing C, then for anys in Fp, P11TDsP11=qDs if and only if P11=CU for U in GO2m+1(Fp).
For Case 2), let k= 2m and s= 1. For all q 6= 0 in Fp, there exists a k×k matrix C so thatCTC=qI. Fixing C, thenP11TP11=qI if and only if P11=CU for U in GO+2m(Fp).
LINDSAY CHILDS
For Case 3), let k = 2m and s be a non-square in Fp. For all q 6= 0 in Fp, there exists a k×k matrix C so that CTDsC = qDs. Fixing C, then P11TDsP11=qDs if and only if P11=CU for U in GO−2m(Fp).
Proof. In Case 1) with k odd, if there exists C so that CTC = qI, then taking determinants gives det(C)2=qk, hence q must be a square.
For the rest, it suffices to find the matrixC in each case.
For Case 1), let q=t2, then C=tI satisfiesCTC =qI.
For Case 2), letq=f2+g2, letQ=
f g
−g f
and letC= diag(Q, Q, . . . , Q).
Then CTC=qI.
For Case 3), let q = f2+g2 and Q as in Case 2). For s a non-square in Fp, find w and x in Fp so that w2 +sx2 = q. (If q is a square, let x = 0, w2 = q; if q is a non-square, let w = 0 and find x so that sx2 = q, possible because the squares have index 2 in F×p.) Then R =
w sx x −w
satisfies RT 1 0
0 s
R =
q 0 0 sq
. Let C = diag(Q, Q, . . . , Q, R). Then
CTDsC=qDs.
Corollary 5.3. Let A be a commutative nilpotent Fp-algebra of dimension n with A3 = 0 and dim(A2) = 1. Suppose the structure matrix of A is Φ = diag(Ds,0)where Ds is k×k and
1) k= 2m+ 1 2) k= 2m, s= 1
3 k= 2m, s is a non-square in Fp.
Then the number of distinct regular subgroups of Affn(Fp) associated to A, and hence the number of Hopf Galois structures on L/K associated to the isomorphism type of A, is
1)
|GLn(Fp)|
(p−12 )· |GO2m+1| · |GLn−1−k| ·pk(n−1−k)+(n−1)
2)
|GLn(Fp)|
(p−1)· |GO+2m| · |GLn−1−k| ·pk(n−1−k)+(n−1)
3)
|GLn(Fp)|
(p−1)· |GO−2m| · |GLn−1−k| ·pk(n−1−k)+(n−1)
The orders of thek×korthogonal groups are polynomials in pof degree (k2 −k)/2 (c.f. [17], p. 72), and the order of GLn(Fp) is a polynomial of degree n2. Hence we have
Corollary 5.4. Let A be a commutative nilpotent Fp-algebra of dimension n with A3 = 0, dim(A2) = 1 and structure matrix of rank k. Let L/K be a
Galois extension with Galois group G ∼= (A,+). Then the number of Hopf Galois structures on L/K of type (A,◦) is a polynomial function of p of degree
n2−(k2−k)/2−(n−k)(n−1)−1 = (2n−k)(k+ 1)
2 −1.
The number of Hopf Galois structures increases with k and is maximal when k=n−1.
6. The cases n= 2,3,4
We compare the counts of Hopf Galois structures in the last section to the number of Hopf Galois structures found by formal group methods in [6]
forn= 2,3.
The case n= 2. Letn= 2, k= 1. Then Φ = (1). ForP to stabilizeT, P =
P11 0 P21 P22
, and the number of choices for each submatrix inP is
|GO1| 1 p p−12
.
Since GO1 ={(1),(−1)}, the size of the stabilizer of the regular subgroup is
2·p·p−1
2 =p(p−1).
The order ofGL2(Fp) is (p2−1)(p2−p). So there arep2−1 distinct regular subgroups in the orbit of the regular subgroup corresponding to Φ.
Since every nilpotent algebra structureA on (F2p,+) hasA3 = 0, we have counted all Hopf Galois structures on a Galois extension with Galois group Cp2.
The case n= 3.
Subcase: k= 1: The matrixP is in the stabilizer of the regular subgroup T corresponding to Φ = diag(1,0) if
P =
P11 0 0 P21 P22 0 P31 P32 P33
,
all submatrices being 1×1. So the number of choices for each entry is
|GO1| 1 1 p |GL1| 1
p p p−12
.
Then |GO1|= 2 and |GL1|=p−1, so the size of the stabilizer Sta(T) is p3(p−1)2,
LINDSAY CHILDS
and the orbit has cardinality
|GL3(Fp)|/|Sta(T)|= (p−1)3(p+ 1).
Subcase: k= 2,s= 1, Φ = diag(1,1): The matrixP is in the stabilizer if P =
P11 0 P21 P22
,
whereP11 is in GO+2. The number of choices for each submatrix is |GO+2| 1
p2 p−1
, and |GO+2|= 2(p−1), so the size of the stabilizer is
2(p−1)2p2.
Subcase: k = 2, s a non-square, Φ = diag(1, s): The matrix P is in the stabilizer if
P =
P11 0 P21 P22
,
whereP11 is in GO−2. The number of choices for each submatrix is |GO−2| 1
p2 p−1
, and |GO−2|= 2(p+ 1), so the size of the stabilizer is
2(p2−1)p2.
The number of regular subgroups corresponding to each case is |GL3| divided by the order of the stabilizer:
Fork= 1, the number of regular subgroups is (p3−1)(p+ 1).
Fork= 2, s= 1, the number of regular subgroups is (p3−1)p(p+ 1)/2.
Fork= 2, s a non-square, the number of regular subgroups is (p3−1)p(p−1)/2.
These agree with the counts found in [6].
The only isomorphism type of nilpotent algebrasA= (F3p,+,·) for which A3 6= 0 is the algebra with dim(A/A2) = 1.
The case n= 4. This case has not previously been looked at.
Forn= 4 there are four subcases:
k= 1. Here
P =
P11 0 0 P21 P22 0 P31 P32 P33
,
whereP22 is 2×2. The number of choices for each submatrix is
|GO1| 1 1 p2 |GL2| 1 p p2 p−12
. So the size of the stabilizer is
2p5(p2−1)(p2−p)
p−1 2
.
k= 2, s= 1: Here P11 is 2×2. The number of choices for each matrix is
|GO+2| 1 1 p2 |GL1| 1 p2 p p−1
. So the size of the stabilizer is
2p5(p−1)3.
k = 2, s a non-square. It is the same as the last case except P11 is in GO−2, so the size of the stabilizer is
2p5(p−1)2(p+ 1).
k= 3. Here
P =
P11 0 P21 P22
where P11 is in GO3, which has order 2p(p2−1), and P22= (q) where q is a square. So the order of the stabilizer is
2p(p2−1)p3p−1 2 .
The number of regular subgroups in each case is the order of GL4(Fp) divided by the orders of the respective stabilizers:
Case number of regular subgroups k= 1 (p2+ 1)(p+ 1)(p3−1) k= 2, s= 1 p(p2+ 1)(p3−1)(p+ 1)2/2 k= 2, s a non-square p(p4−1)(p3−1)/2
k= 3 p2(p4−1)(p3−1)
LINDSAY CHILDS
The total number of Hopf Galois structure exceeds p9. Note that the degrees of the polynomials in each case agree with Corollary5.4.
The Hopf Galois structure. Given a Galois extension L/K with Galois group G ∼= Fnp, if the commutative nilpotent algebra A with dim(A2) = 1, A3 = 0 has diagonal structure matrix Φ = diag(d1, . . . , dk,0, . . . ,), then the regular subgroup T corresponding toD acts onGL by
λ◦(r)(et) =er◦t=ew where
w=r+t+
k
X
i=1
ritidi
! xn, and λ(G) conjugatesT by
λ(t)λ◦(r)λ(−t) =λ◦(r−r·t) =λ◦ r−
k
X
i=1
ritidi)xn
! . 7. The Galois correspondence ratio
LetAbe a commutative Fp-algebra of dimensionnwithA3 = 0, yielding the skew brace (A,+,◦). In [8] (generalized in [9] and [10]) we showed that for a Galois extensionL/K with Galois group (A,+) and anH-Hopf Galois structure of type (A,◦), the image of the Galois correspondence for the Hopf Galois structure is in bijective correspondence with the ideals of A. Thus the Galois correspondence ratio for the Hopf Galois structure is
GC(L/K,(A,+), H) = |{ideals of A}|
|{subgroups of (A,+)}|. We have
Proposition 7.1. Let A be as in Corollary 5.3 with a non-zero structure matrix Ds of rank k ≥ 1. Let L/K be a Galois extension with Galois group G∼= (A,+) and a Hopf Galois structure associated to the skew brace (A,+,◦). Then
GC(L/K,(A,+), H) =O 1
p(n−1)/2
for nodd;
=O 1
pn/2
for neven.
Proof. The denominator of GC(L/K,(A,+), H) is equal to the number of subspaces of Fnp, a known quantity. So to estimate this ratio, we need to estimate the number of ideals ofA.
The algebra A = (A, n, k) has basis (x1, x2, . . . , xn) where x2i = xn for i = 1,2, . . . , k−1, x2k = s 6= 0 and x2i = 0 for k < i ≤ n. Viewing A as a vector space with basis x1, . . . xn, we know (c.f [11], Section 1) that the
number of subspaces of Fnp of dimension k is ≥ pk(n−k) and has order of magnitude =pk(n−k).
We can count the total number of subspaces ofAby viewing the subspaces of A as row spaces of n×n matrices with entries in Fp and counting the number of parameters of all possible reduced row echelon forms of those n×n matrices. So let R = (c1, c2, . . . , cr) denote the general reduced row echelon form of rankrwithrnon-zero rows and pivots in columns numbered c1, c2, . . . , cr. Then the number nR of Fp-parameters in the matrix R = (c1, c2, . . . , cr), (counting row by row from the top) is equal to
(c2−c1−1)+2(c3−c2−1)+. . .+. . .+(r−1)(cr−cr−1−1)+r(n+1−cr−1).
So the dimension of the subspace defined by the matrixR is mR=pnR
=pc2−c1−1·p2(c3−c2−1)· · · · ·p(r−1)(cr−cr−1−1)·pr(n+1−cr−1). The largestmRcan be is if (c1, c2, . . . , cr) = (1,2, . . . , r), so that the product reduces to the single term pr = pr(n+1−r−1). Thus forn even, the number s(Fnp)of subspaces of Fnp is a polynomial in p with a unique highest degree term, when r =n/2, namely pn2/4. For n odd, s(Fnp) is a polynomial in p with two equal highest degree terms, when r= (n−1)/2 or r = (n+ 1)/2, namely p(n2−1)/4. Thus the leading term ofs(Fnp) for nodd is = 2p(n2−1)/4. Now we estimate the number of ideals ofA, assuming that inA,x21=dxn
withd6= 0. The key fact is that if a matrixR represents a subspace which is an ideal and contains an element x = x1 +a2x2 +. . .+anxn, then it also contains x1x=dxn. SoR must contain a row (0,0, . . . ,0,1). Thus the matrices
R= (1,2,3, . . . , r)
which give the largest number of parameters do not represent ideals, while the matrices
RI= (1,2,3, . . . , r, n)
do represent ideals, but have r fewer parameters than R does. Also R0 = (2,3, . . . r) represents an ideal ifx22 = 0, but R0 hasn−r fewer parameters thanR. In particular, fornodd, the matrixRI giving the most parameters is
RI =
1,2,3, . . . ,n−1 2 , n
,
namely, (n2−1)/4 parameters, and for n even, the matrix RI giving the most parameters is
RI =
1,2,3, . . . ,n−1 2 , n
, namely n2/4.
LINDSAY CHILDS
Thus in the case of an A closest to the trivial algebra A = (Fnp,+), the ratio
#{ideals ofA}/#{subspaces ofA}=O(1/(p(n−1)/2)) orO(1/pn/2)
fornodd, resp. even.
If A has x2i = sixn for si 6= 0 for i = 1, . . . , d, the number of subspaces that are ideals decreases asdincreases, to the point where ifd=n−1, then the ideals ofA are the subspaces ofAthat contain xn. Then the number of non-zero ideals of Ais equal to the number of subspaces of Fn−1p .
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(Lindsay Childs)Department of Mathematics and Statistics, University at Al- bany, Albany, NY 12222, USA
This paper is available via http://nyjm.albany.edu/j/2019/25-58.html.
